6.042/18.062J Mathematics for Computer Science
Tom Leighton and Marten van Dijk
September 28, 2010
Problem Set 4
Problem 1. [15 points] Let G = (V, E) be a graph. A matching in G is a set M ⊂ E
such that no two edges in M are incident on a common vertex.
Let M1 , M2 be two matchings of G. Consider the new graph G� = (V, M1 ∪ M2 ) (i.e. on the
same vertex set, whose edges consist of all the edges that appear in either M1 or M2 ). Show
that G� is bipartite.
Helpful definition: A connected component is a subgraph of a graph consisting of some vertex
and every node and edge that is connected to that vertex.
Problem 2. [20 points] Let G = (V, E) be a graph. Recall that the degree of a vertex
v ∈ V , denoted dv , is the number of vertices w such that there is an edge between v and
w.
(a) [10 pts] Prove that
2|E| =
�
dv .
v∈V
(b) [5 pts] At a 6.042 ice cream study session (where the ice cream is plentiful and it helps
you study too) 111 students showed up. During the session, some students shook hands with
each other (everybody being happy and content with the ice-cream and all). Turns out that
the University of Chicago did another spectacular study here, and counted that each student
shook hands with exactly 17 other students. Can you debunk this too?
(c) [5 pts] And on a more dull note, how many edges does Kn , the complete graph on n
vertices, have?
Problem 3. [15 points] Two graphs are isomorphic if they are the same up to a relabeling
of their vertices (see Definition 5.1.3 in the book). A property of a graph is said to be
preserved under isomorphism if whenever G has that property, every graph isomorphic to G
also has that property. For example, the property of having five vertices is preserved under
isomorphism: if G has five vertices then every graph isomorphic to G also has five vertices.
(a) [5 pts] Some properties of a simple graph, G, are described below. Which of these
properties is preserved under isomorphism?
2
Problem Set 4
1. G has an even number of vertices.
2. None of the vertices of G is an even integer.
3. G has a vertex of degree 3.
4. G has exactly one vertex of degree 3.
(b) [10 pts] Determine which among the four graphs pictured in the Figures are isomorphic.
If two of these graphs are isomorphic, describe an isomorphism between them. If they are not,
give a property that is preserved under isomorphism such that one graph has the property,
but the other does not. For at least one of the properties you choose, prove that it is indeed
preserved under isomorphism (you only need prove one of them).
1
1
6
6
5
8
2
9
10
5
7
10
7
4
9
8
4
3
3
(a) G1
(b) G2
1
1
9
2
2
6
3
8
5
9
7
2
10
10
8
4
7
6
5
(c) G3
4
3
(d) G4
Figure 1: Which graphs are isomorphic?
Problem 4. [15 points] Recall that a coloring of a simple graph is an assignment of a
color to each vertex such that no two adjacent vertices have the same color. A k-coloring
is a coloring that uses at most k colors.
False Claim. Let G be a (simple) graph with maximum degree at most k. If G also has a
vertex of degree less than k, then G is k-colorable.
(a) [5 pts] Give a counterexample to the False Claim when k = 2.
(b) [10 pts] Consider the following proof of the False Claim:
Problem Set 4
3
Proof. Proof by induction on the number n of vertices:
Induction hypothesis: P (n) is defined to be: Let G be a graph with n vertices and
maximum degree a most k. If G also has a vertex of degree less than k, then G is k-colorable.
Base case: (n=1) G has only one vertex and so is 1-colorable. So P (1) holds.
Inductive step:
We may assume P (n). To prove P (n + 1), let Gn+1 be a graph with n + 1 vertices and
maximum degree at most k. Also, suppose Gn+1 has a vertex, v, of degree less than k. We
need only prove that Gn+1 is k-colorable.
To do this, first remove the vertex v to produce a graph, Gn , with n vertices. Removing v
reduces the degree of all vertices adjacent to v by 1. So in Gn , each of these vertices has
degree less than k. Also the maximum degree of Gn remains at most k. So Gn satisfies the
conditions of the induction hypothesis P (n). We conclude that Gn is k-colorable.
Now a k-coloring of Gn gives a coloring of all
degree less than k, there will be fewer than k
among the k possible colors, there will be a
and this color can be assigned to v to form a
the vertices of Gn+1 , except for v. Since v has
colors assigned to the nodes adjacent to v. So
color not used to color these adjacent nodes,
k-coloring of Gn+1 .
Identify the exact sentence where the proof goes wrong.
Problem 5. [15 points] Prove or disprove the following claim: for some n ≥ 3 (n boys
and n girls, for a total of 2n people), there exists a set of boys’ and girls’ preferences such
that every dating arrangement is stable.
Problem 6. [20 points]
Let (s1 , s2 , ..., sn ) be an arbitrarily distributed sequence of the number 1, 2, ..., n − 1, n. For
instance, for n = 5, one arbitrary sequence could be (5, 3, 4, 2, 1).
Define the graph G=(V,E) as follows:
1. V = {v1 , v2 , ..., vn }
2. e = (vi , vj ) ∈ E if either:
(a) j = i + 1, for 1 ≤ i ≤ n − 1
(b) i = sk , and j = sk+1 for 1 <= k <= n − 1
(a) [10 pts] Prove that this graph is 4-colorable for any (s1 , s2 , ..., sn ).
Hint: First show that a line graph is 2-colorable. Note that a line graph is defined as follows:
The n-node graph containing n − 1 edges in sequence is known as the line graph Ln .
(b) [10 pts] Suppose (s1 , s2 , ..., sn ) = (1, a1 , 3, a2 , 5, a3 , ...) where a1 , a2.... is an arbitrary
distributed sequence of the even numbers in 1, ..., n − 1. Prove that the resulting graph is
2-colorable.
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6.042J / 18.062J Mathematics for Computer Science
Fall 2010
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